Modulation equations getdocf6a1. 198KB Jun 04 2011 12:05:13 AM

the process η ǫ t t ∈[0,T ] converges as ǫ goes to zero to a Gaussian process ηt satisfying a linear equation with additive noise. More precisely, for any T 0, lim ǫ→0 E sup t ∈[0,T ∧τ ǫ α ] |η ǫ t − ηt| 2 L 2 = 0. The conclusion of Theorem 3.1 will be obtained in two steps. The first step consists in estimating the modulation parameters obtained in Theorem 2.1, in terms of η ǫ , using the equations for those parameters; then the convergence of η ǫ as ǫ tends to zero is proved. In the next section, a slight change in the modulation parameters will be performed, in order to precise the asymptotic behavior in time of the limit process η. It will indeed be proved that part of it is actually an Ornstein- Uhlenbeck process. From now on, we assume that α is fixed and sufficiently small, so that the conclusion of Theorem 2.1 holds, and we denote τ ǫ α by τ ǫ .

3.1 Modulation equations

Since we know that the modulation parameters x ǫ t and c ǫ t are semi-martingale processes adapted to the filtration generated by W t t ≥0 , we may a priori write the stochastic evolution equations for those parameters in the form ¨ d x ǫ = c ǫ d t + ǫ y ǫ d t + ǫz ǫ , dW d c ǫ = ǫa ǫ d t + ǫb ǫ , dW 3.1 where y ǫ and a ǫ are real valued adapted processes with a.s. locally integrable paths on [0, τ ǫ , and b ǫ , z ǫ are predictable processes with paths a.s. in L 2 l oc 0, τ ǫ ; L 2 R. We then proceed as in [5] : the Itô-Wentzell Formula applied to u ǫ t, x + x ǫ t, together with equation 1.1 for u ǫ and the first equation of 3.1 for x ǫ give a stochastic evolution equation for u ǫ t, x + x ǫ . Note that one may use the Itô-Wentzell formula given in [17]. Indeed, it is easily checked that the process u ǫ stopped at τ ǫ satisfies the assumptions of Theorem 1.1 in [17]. The process x ǫ t does not readily satisfy the required assumptions, since the processes y ǫ and z ǫ , φe k are not bounded on Ω × R + , even when they are stopped at τ ǫ . However, one may use a cut-off for y ǫ and z ǫ , φe k , apply the formula of [17] and then remove the cut-off, noticing that each term in the equation converges a.s., at least as a distribution. On the other hand, the standard Itô Formula together with the second equation of 3.1 for c ǫ give an equation for the evolution of ϕ c ǫ t . Replacing then ϕ c ǫ t + ǫη ǫ t, x for u ǫ t, x + x ǫ t in the first equation leads to the following stochastic equation for the evolution of η ǫ t : d η ǫ = ∂ x L c η ǫ d t + y ǫ ∂ x ϕ c ǫ − a ǫ ∂ c ϕ c ǫ d t − ∂ x ϕ c ǫ − ϕ c η ǫ d t +c ǫ − c + ǫ y ǫ ∂ x η ǫ d t − ǫ 2 ∂ x η ǫ 2 d t + ϕ c ǫ T x ǫ dW +∂ x ϕ c ǫ z ǫ , dW − ∂ c ϕ c ǫ b ǫ , dW + ǫη ǫ T x ǫ dW + ǫ∂ x η ǫ z ǫ , dW + ǫ 2 ∂ 2 x ϕ c ǫ |φ ∗ z ǫ | 2 L 2 d t − ǫ 2 ∂ 2 c ϕ c ǫ |φ ∗ b ǫ | 2 L 2 d t + ǫ X l ∈N ∂ x ϕ c ǫ T x ǫ φe l z ǫ , φe l d t + 1 2 ǫ 2 ∂ 2 x η ǫ |φ ∗ z ǫ | 2 L 2 d t + ǫ 2 X l ∈N ∂ x η ǫ T x ǫ φe l z ǫ , φe l d t 3.2 1734 where L c is defined in 2.7. Now, taking the L 2 - inner product of equation 3.2 with ϕ c , on the one hand, and with ∂ x ϕ c on the other hand, then using the orthogonality conditions 2.5 and the fact that L c ∂ x ϕ c = 0, and finally identifying the drift parts and the martingale parts of each of the resulting equations lead to the same kind of system that we previously obtained in [5]; namely, setting Y ǫ t = ‚ y ǫ t a ǫ t Œ and Z ǫ l t = ‚ z ǫ , φe l b ǫ , φe l Œ then one gets for the drift parts A ǫ tY ǫ t = G ǫ t 3.3 where A ǫ t = ‚ ∂ x ϕ c ǫ + ǫ∂ x η ǫ , ∂ x ϕ c −∂ c ϕ c ǫ , ∂ x ϕ c −∂ x ϕ c ǫ , ϕ c ∂ c ϕ c ǫ , ϕ c Œ 3.4 and G ǫ t = ‚ G ǫ 1 t G ǫ 2 t Œ , with G ǫ 1 t = η ǫ , L c ∂ 2 x ϕ c + c ǫ − c η ǫ , ∂ 2 x ϕ c + ǫ 2 ∂ x η ǫ 2 , ∂ x ϕ c +∂ x ϕ c ǫ − ϕ c η ǫ , ∂ x ϕ c − ǫ 2 ∂ 2 x ϕ c ǫ , ∂ x ϕ c |φ ∗ z ǫ | 2 L 2 + ǫ 2 ∂ 2 c ϕ c ǫ , ∂ x ϕ c |φ ∗ b ǫ | 2 L 2 − ǫ X l ∈N z ǫ , φe l ∂ x ϕ c ǫ T x ǫ φe l , ∂ x ϕ c + 1 2 ǫ 2 η ǫ , ∂ 3 x ϕ c |φ ∗ z ǫ | 2 L 2 − ǫ 2 X l ∈N ∂ x η ǫ T x ǫ φe l , ∂ x ϕ c z ǫ , φe l 3.5 and G ǫ 2 t = − ǫ 2 ∂ x η ǫ 2 , ϕ c − ∂ x ϕ c ǫ − ϕ c η ǫ , ϕ c + ǫ 2 ∂ 2 x ϕ c ǫ , ϕ c |φ ∗ z ǫ | 2 L 2 − ǫ 2 ∂ 2 c ϕ c ǫ , ϕ c |φ ∗ b ǫ | 2 L 2 + ǫ X z ǫ , φe l ∂ x ϕ c ǫ T x ǫ φe l , ϕ c + ǫ 2 2 η ǫ , ∂ 2 x ϕ c |φ ∗ z ǫ | 2 L 2 + ǫ 2 X l ∈N ∂ x η ǫ T x ǫ φe l , ϕ c z ǫ , φe l ; 3.6 note that A ǫ t = A + O|c ǫ − c | + kǫη ǫ k 1 , a.s. for t ≤ τ ǫ with A = ‚ |∂ x ϕ c | 2 L 2 ϕ c , ∂ c ϕ c Œ and O |c ǫ − c | + kη ǫ k 1 is uniform in ǫ, t and ω as long as t ≤ τ ǫ . Concerning the martingale parts, one gets the equation A ǫ tZ ǫ l t = F ǫ l t, ∀l ∈ N 3.7 with F ǫ t = ‚ −ϕ c ǫ + ǫη ǫ T x ǫ φe l , ∂ x ϕ c ϕ c ǫ + ǫη ǫ T x ǫ φe l , ϕ c . Œ 3.8 1735 Proposition 3.2. Under the above assumptions, there is a constant α 1 0, such that if α ≤ α 1 , then |φ ∗ z ǫ t| L 2 + |φ ∗ b ǫ | L 2 ≤ C 1 |k| L 2 , a.s. for t ≤ τ ǫ 3.9 and |a ǫ t| + | y ǫ t| ≤ C 2 |η ǫ t| L 2 + ǫC 3 , a.s. for t ≤ τ ǫ 3.10 for some constants C 1 , C 2 , C 3 , depending only on α and c , and for any ǫ ≤ ǫ . Proof. The proof is exactly the same as the proof of Corollary 4.3 in [5], once noticed that, a.s. for t ≤ τ ǫ , X l ∈N |F ǫ l t| 2 ≤ C X l ∈N |ϕ c ǫ + ǫη ǫ T x ǫ φe l | 2 L 2 ≤ C X l Z R ϕ c ǫ + ǫη ǫ 2 x[T x ǫ k ∗ e l ] 2 xd x ≤ Z R ϕ c ǫ + ǫη ǫ 2 x X l T x ǫ kx − ., e l 2 d x ≤ C Z R ϕ c ǫ + ǫη ǫ 2 x|T x ǫ kx − .| 2 L 2 d x ≤ C|k| 2 L 2 |ϕ c ǫ + ǫη ǫ | 2 L 2 ≤ C|k| 2 L 2 where we have used the Parseval equality in the fourth line. ƒ

3.2 Convergence of